How can you find the maximal volume of a rectangular box inscribed in a sphere?
How can you find the maximal volume of a rectangular box inscribed in a sphere?
Equations (4) and (5) are partial derivatives and have to be equated to zero to maximize the volume of the rectangular box. Hence, the maximal volume of a rectangular box inside the sphere is 8r3/3√3.
How do you find the volume of a cube inside a sphere?
If we let the edge length of the cube be a, then the largest diagonal of the cube will be equal to a√3 by the Pythagorean Theorem, and this will also be the diameter. From the formula for the volume of a sphere given the diameter (16πd3), the area of the sphere is equal to π6×(a√32)3=πa3√316.
Why is the volume of a box 8xyz?
Clearly the box will have the greatest volume if each of its corners touch the ellipse. Let one of the corners (x, y, z) be in the positive octant, then the box has corners (±x, ±y, ±z) and its volume is V = 8xyz.
What is the volume of the sphere?
The general formula for the volume of sphere in terms of its radius is given as V = (4/3) π r3.
How do you find the dimensions of a rectangular box that has the largest volume and surface area of 56 square units?
The maximum volume is 56√219≈28.5 cubic units. Let x , y , and z be the dimensions of the box. Then V=xyz and S=2xy+2xz+2yz are the volume and surface areas of the box.
How do you find the maximum volume of a rectangular prism?
To find the volume of a rectangular prism, multiply its 3 dimensions: length x width x height. The volume is expressed in cubic units.
What is the volume of a box?
To find the volume of a box, simply multiply length, width, and height — and you’re good to go! For example, if a box is 5×7×2 cm, then the volume of a box is 70 cubic centimeters.
Which of the following is the volume of the largest sphere that can fit inside of a cube?
Notice that the largest possible sphere that can fit inside the cube is the inscribed sphere, which has radius 12s. Using the volume formula for a sphere, we find that Vsphere=43πr3=43πs38=π6s3.
What is volume of a rectangle?
The formula for finding the volume of a rectangular prism is the following: Volume = Length * Height * Width, or V = L * H * W.
Which is the maximum rectangular box inscribed in a sphere?
The maximum rectangular box inscribed in a sphere is a cube. The diagonal of the cube is the diameter of the sphere. Diagonal of cube with side s is √ (3) √ (s²) = s √3 = diameter; therefore s = (diameter / √3) units.
What is the maximum volume of a sphere?
V = 4/3*pi*r^3, where r is the radius and V is the volume. Since the diameter is 2r, then half the diameter is r; half of 20cm is 10cm. We now have the following equation for the volume of this sphere: So the maximum volume of this sphere is 4,176.2 cubic centimeters.
How to calculate the volume of a box?
Then the volume of the box is 8 x y z. The sphere of radius a is given by x 2 + y 2 + z 2 = a 2. g ( x, y, z) = 0 where g ( x, y, z) = x 2 + y 2 + z 2 − a 2.
How to calculate the volume of a cube?
√ (3) √ (s²) = s √3 = diameter; therefore s = (diameter / √3) units. The volume of the cube is D³ / (3 √3) cubic units. You can solve it with simple logic applying elementary geometry or use a more complex calculus solution applying derivatives.
